Kyungpook Mathematical Journal 2021; 61(3): 495-512
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
Value Distribution of L-functions and a Question of Chung-Chun Yang
Xiao-Min Li*, Qian-Qian Yuan, Hong-Xun Yi
Department of Mathematics, Ocean University of China, Qingdao, Shandong 266100, P. R. China
e-mail : firstname.lastname@example.org and email@example.com
Department of Mathematics, Shandong University, Jinan, Shandong 250199, P. R. China
e-mail : firstname.lastname@example.org
Received: February 22, 2017; Revised: May 2, 2021; Accepted: May 18, 2021
We study the value distribution theory of L-functions and completely resolve a question from Yang . This question is related to L-functions sharing three finite values with meromorphic functions. The main result in this paper extends corresponding results from Li .
Keywords: Nevanlinna theory, Meromorphic functions, L-functions, Shared Values, Uniqueness theorems.
1. Introduction and Main Results
Throughout this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We assume that the reader is familiar with the basic notions and results in the Nevanlinna theory, which can be found, for example, in [4, 9, 18, 19]. It will be convenient to let
This paper concerns the question of how an L-function is uniquely determined in terms of the pre-images of complex values in the extended complex plane, or sharing values. We refer the reader to the monograph  for a detailed discussion on the topic and related works. Throughout the paper, an L-function always means an L-function L in the Selberg class, which includes the Riemann zeta function
(i) Ramanujan hypothesis.
(ii) Analytic continuation. There is a nonnegative integer
(iii) Functional equation.
with positive real numbers
(iv) Euler product hypothesis.
We first recall the following result due to Steuding , which actually holds without the Euler product hypothesis:
Theorem A. ([17, p.152]) If two L-functions
Remark 1.1. In 2016, Hu-Li  pointed out that Theorem A is false when
Theorem A implies that two L-functions with
Theorem B.() Let
Remark 1.2. The number "two" in Theorem B is the best possible, as shown by the above example with
By Theorem B we can get the following result:
Corollary A.() Let
In a communication to Professor Li, Yang asked the following question:
Question A.() If
Remark 1.3. By taking
Next we consider the first, the second and the fourth Painlevé equations given respectively by
In 2007, Lin-Tohge  obtained some results similar to Theorem B. Indeed, Lin-Tohge  studied some shared-value properties of the first, the second and the fourth Painlevé transcendents by applying their distinctive value distribution, and proved the following results:
Theorem C.([13, Theorem 1]) Let ω be an arbitrary nonconstant solution of one of the equations (PI), (PII) and (PIV), and let
Theorem D.([13, Theorem 2]) Let ω be an arbitrary solution of (PI) and
Theorem F.([13, Theorem 3]) Let ω be an arbitrary solution of (PI). Then there does not exist a pair of two finite values
Regarding Theorem B, one may ask, what can be said about the conclusion of Theorem B if we remove the assumption "
Theorem 1.1. Let
By Theorem 1.1 we get the following result:
Corollary 1.2. If
As a special case of Corollary 1.2, we give the following result which completely resolves Question A:
Corollary 1.3. If
In the same manner as in the proof of Theorem 1.1, we can get the following result by Lemma 2.10 in Section 2 of the present paper:
Theorem 1.4. Let
Throughout this paper, we will apply Nevanlinna theory to prove the main result in this paper.
In this section, we will give some important lemmas to prove the main result of the present paper. For convenience in stating the following first result from Gundersen , we shall use the following notation: we shall let
of distinct pairs of integers that satisfy
Lemma 2.1.([3, Corollary 2]) Let
The following result is due to Mokhon-ko :
Lemma 2.2.(Valiron-Mokhon-ko lemma, ) Let
We also need the following result due to Lahiri-Sarkar :
Lemma 2.3.([8, Lemma 6]) Let
The following result is from Gundersen :
Lemma 2.4.([2, Theorem 3]) Suppose that
Lemma 2.5.([20, proof of Lemma 4]) Let
Lemma 2.6.([21, Lemma 6]) Let
For introducing the following result, we first give the following notation (cf.): Let
The following lemma is essentially due to Zhang :
Lemma 2.7.([21, proof of Theorem 1 and Theorem 2]) Let
Here γ is a nonconstant entire function,
Lemma 2.8. () Let
Finally we prove the following result which plays an important role in proving the main results of this paper:
Lemma 2.9. Let
By Lemma 2.4 we have
By (2.4), Lemma 2.5 and the assumption of Lemma 2.9 we have
By integrating two sides of (2.14) we can get
which together with (2.8) gives
and so we have
By (2.28) and Lemma 2.5 we deduce
In the same manner as above we get
Lemma 2.10.() Let
such that, for every positive integer
3. Proof of Theorem 1.1.
First of all, we denote by
which together with the definition of the order of a meromorphic function implies that
By noting that
On the other hand, by the assumption that
Now we set
By (3.9) we consider the following two cases:
Case 1. Suppose that there exists a subset
Next we prove
Case 2. Suppose that at most there exists a subset
Then, by (3.15) we have
Noting the assumption that
We discuss the following two subcases:
Subcase 2.1. Suppose that
Subcase 2.1.1. Suppose that
By (3.20) and Lemma 2.2 we have
By (3.6), (3.21) and the definition of the order of a meromorphic function we have
which contradicts (3.1).
Subcase 2.1.2. Suppose that
By (3.6), (3.23) and the definition of the order of a meromorphic function we have
which contradicts (3.1).
We consider the following two subcases:
Subcase 2.2.1. Suppose that
By noting that
where γ is a nonconstant entire function,
which contradicts (3.1).
Subcase 2.2.2. Suppose that
By (3.34) we deduce that
The authors wish to express their thanks to the referee for his/her valuable suggestions and comments.
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